Taylor-Accelerated Neural Network Interpolation Operators on Irregular Grids with Higher Order Approximation

Abstract: In this paper, a new class of \emph{Taylor-accelerated neural network interpolation operators} is introduced on quasi-uniform irregular grids. These operators improve existing neural network interpolation operators by incorporating Taylor polynomials at the sampling nodes, thereby exploiting the higher smoothness of the target function. The proposed operators are shown to be well defined, uniformly bounded, and to satisfy an exact interpolation property at the grid points. In addition, polynomial reproduction up to a prescribed degree is established. Direct approximation estimates are derived in terms of higher-order moduli of smoothness, yielding enhanced convergence rates for sufficiently smooth functions. Numerical experiments are presented to support the theoretical analysis and to demonstrate the significant accuracy improvement achieved through the Taylor-accelerated construction. In particular, higher-order convergence on irregular grids is obtained, and the proposed approach outperforms existing neural network interpolation operators on irregular partitions, including Lagrange-based schemes.